3.186 \(\int x^3 \cot (a+i \log (x)) \, dx\)

Optimal. Leaf size=49 \[ -i e^{2 i a} x^2-i e^{4 i a} \log \left (-x^2+e^{2 i a}\right )-\frac {i x^4}{4} \]

[Out]

-I*exp(2*I*a)*x^2-1/4*I*x^4-I*exp(4*I*a)*ln(exp(2*I*a)-x^2)

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Rubi [F]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^3 \cot (a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*Cot[a + I*Log[x]],x]

[Out]

Defer[Int][x^3*Cot[a + I*Log[x]], x]

Rubi steps

\begin {align*} \int x^3 \cot (a+i \log (x)) \, dx &=\int x^3 \cot (a+i \log (x)) \, dx\\ \end {align*}

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Mathematica [B]  time = 0.04, size = 137, normalized size = 2.80 \[ x^2 \sin (2 a)-i x^2 \cos (2 a)-\cos (4 a) \tan ^{-1}\left (\frac {\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-i \sin (4 a) \tan ^{-1}\left (\frac {\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-\frac {1}{2} i \cos (4 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )+\frac {1}{2} \sin (4 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )-\frac {i x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Cot[a + I*Log[x]],x]

[Out]

(-1/4*I)*x^4 - I*x^2*Cos[2*a] - ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Cos[4*a] - (I/2)*Cos[4*a]*L
og[1 + x^4 - 2*x^2*Cos[2*a]] + x^2*Sin[2*a] - I*ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Sin[4*a] +
(Log[1 + x^4 - 2*x^2*Cos[2*a]]*Sin[4*a])/2

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fricas [A]  time = 0.66, size = 32, normalized size = 0.65 \[ -\frac {1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} - i \, e^{\left (4 i \, a\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cot(a+I*log(x)),x, algorithm="fricas")

[Out]

-1/4*I*x^4 - I*x^2*e^(2*I*a) - I*e^(4*I*a)*log(x^2 - e^(2*I*a))

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giac [A]  time = 0.55, size = 50, normalized size = 1.02 \[ -\frac {1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} + \frac {1}{2} \, \pi e^{\left (4 i \, a\right )} - i \, e^{\left (4 i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, e^{\left (4 i \, a\right )} \log \left (-x + e^{\left (i \, a\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cot(a+I*log(x)),x, algorithm="giac")

[Out]

-1/4*I*x^4 - I*x^2*e^(2*I*a) + 1/2*pi*e^(4*I*a) - I*e^(4*I*a)*log(x + e^(I*a)) - I*e^(4*I*a)*log(-x + e^(I*a))

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maple [A]  time = 0.07, size = 39, normalized size = 0.80 \[ -i {\mathrm e}^{2 i a} x^{2}-\frac {i x^{4}}{4}-i {\mathrm e}^{4 i a} \ln \left ({\mathrm e}^{2 i a}-x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cot(a+I*ln(x)),x)

[Out]

-I*exp(2*I*a)*x^2-1/4*I*x^4-I*exp(4*I*a)*ln(exp(2*I*a)-x^2)

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maxima [B]  time = 0.34, size = 136, normalized size = 2.78 \[ -\frac {1}{4} i \, x^{4} - x^{2} {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} + \frac {1}{4} \, {\left (4 \, \cos \left (4 \, a\right ) + 4 i \, \sin \left (4 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) - \frac {1}{4} \, {\left (4 \, \cos \left (4 \, a\right ) + 4 i \, \sin \left (4 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right ) - \frac {1}{2} \, {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - \frac {1}{2} \, {\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cot(a+I*log(x)),x, algorithm="maxima")

[Out]

-1/4*I*x^4 - x^2*(I*cos(2*a) - sin(2*a)) + 1/4*(4*cos(4*a) + 4*I*sin(4*a))*arctan2(sin(a), x + cos(a)) - 1/4*(
4*cos(4*a) + 4*I*sin(4*a))*arctan2(sin(a), x - cos(a)) - 1/2*(I*cos(4*a) - sin(4*a))*log(x^2 + 2*x*cos(a) + co
s(a)^2 + sin(a)^2) - 1/2*(I*cos(4*a) - sin(4*a))*log(x^2 - 2*x*cos(a) + cos(a)^2 + sin(a)^2)

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mupad [B]  time = 2.22, size = 38, normalized size = 0.78 \[ -x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,1{}\mathrm {i}-\ln \left (x^2-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,1{}\mathrm {i}-\frac {x^4\,1{}\mathrm {i}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cot(a + log(x)*1i),x)

[Out]

- x^2*exp(a*2i)*1i - log(x^2 - exp(a*2i))*exp(a*4i)*1i - (x^4*1i)/4

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sympy [A]  time = 0.22, size = 39, normalized size = 0.80 \[ - \frac {i x^{4}}{4} - i x^{2} e^{2 i a} - i e^{4 i a} \log {\left (x^{2} - e^{2 i a} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*cot(a+I*ln(x)),x)

[Out]

-I*x**4/4 - I*x**2*exp(2*I*a) - I*exp(4*I*a)*log(x**2 - exp(2*I*a))

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